مسائل تکمیل در ماتریس‌های عملگر

Completion problems for operator matrices are concerned with the question of whether a partially specified operator matrix can be completed to form an operator of a desired type. The research devoted to this topic provides an excellent means to investigate the structure of operators. This book provides an overview of completion problems dealing with completions to different types of operators and can be considered as a natural extension of classical results concerned with matrix completions. The book assumes some basic familiarity with functional analysis and operator theory. It will be useful for graduate students and researchers interested in operator theory and the problem of matrix completions. Cover 1 Title page 4 Contents 6 Preface 8 Notation 12 Chapter 1. Completion problems for upper triangular operator matrices 16 1.1. Completions of upper triangular operator matrices to injective and left invertible operator 17 1.2. Completions of upper triangular operator matrices to invertible and right invertible operator 22 1.3. Completions of upper triangular operator matrices to regular operator 28 1.4. Completions of upper triangular operator matrices to Drazin and generalized Drazin invertible operator 34 1.5. Completions of upper triangular operator matrices to Fredholm operator 43 1.6. Completions of upper triangular operator matrices to generalized left and right Weyl operator 51 1.7. Completions of upper triangular operator matrices to Browder operator 59 1.8. Completions of upper triangular matrices to Kato nonsingular operator 67 1.9. Fredholm consistency of upper triangular operator matrices 78 Chapter 2. Invertibility of an operator A+CX 90 2.1. Auxiliary results 90 2.2. Invertibility of an operator A+CX 92 2.3. Injectivity of an operator A+CX 96 2.4. Surjectivity of an operator A+CX 99 2.5. Right and left invertibility of an operator A+CX 101 2.6. Fredholmness of an operator A+CX 103 2.7. Dense range of an operator A+CX 108 2.8. Closed range of an operator A+CX 110 Chapter 3. Completions of operator matrices M_{X} 116 3.1. Completions of operator matrices M_{X} to invertible operator 117 3.2. Completions of M_{X} to injective and nondense range operator 122 3.3. Completions of operator matrices M_{X} to right and left invertible operator 128 3.4. Completions of operator matrices M_{X} to semi-Fredholm operator 130 3.5. Completions of operator matrices M_{X} to Fredholm operator 135 3.6. Completions of operator matrices M_{X} to closed range operator 137 Chapter 4. Completions of operator matrices M_{(X,Y)} 140 4.1. Completions of operator matrices M_{(X,Y)} to invertibility, right and left invertibility 140 4.2. Consistency of operator matrices M_{(X,Y)} in invertibility 143 4.3. Consistency of operator matrices M_{(X,Y)} in Fredholmness 148 Chapter 5. Applications of completion results 154 5.1. Invertibility of linear combinations of operators 154 5.2. Injectivity, right and left invertibility of the sum of two operators 163 5.3. Fredholmness of the sum of two operators 166 Bibliography 174 Index 184 Back Cover 187